| L(s) = 1 | + 1.56·3-s − 5-s − 0.438·7-s − 0.561·9-s + 11-s + 7.12·13-s − 1.56·15-s + 4.68·17-s − 5.56·19-s − 0.684·21-s + 7.12·23-s + 25-s − 5.56·27-s + 4.43·29-s + 5.56·31-s + 1.56·33-s + 0.438·35-s + 11.5·37-s + 11.1·39-s + 4.24·41-s − 5.12·43-s + 0.561·45-s − 13.3·47-s − 6.80·49-s + 7.31·51-s − 2.68·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s − 0.447·5-s − 0.165·7-s − 0.187·9-s + 0.301·11-s + 1.97·13-s − 0.403·15-s + 1.13·17-s − 1.27·19-s − 0.149·21-s + 1.48·23-s + 0.200·25-s − 1.07·27-s + 0.824·29-s + 0.998·31-s + 0.271·33-s + 0.0741·35-s + 1.90·37-s + 1.78·39-s + 0.663·41-s − 0.781·43-s + 0.0837·45-s − 1.95·47-s − 0.972·49-s + 1.02·51-s − 0.368·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072061088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072061088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 13 | \( 1 - 7.12T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03074935626406960154103357609, −9.099967237966268641525759865103, −8.360088928861532647193462840980, −7.999164874241700356366186399819, −6.65172095489332773595196796393, −5.97776496067737936530168155956, −4.57041588382976734943465349390, −3.54836043949067546959013704476, −2.87859095198752881762150845620, −1.23189008634241684855923277270,
1.23189008634241684855923277270, 2.87859095198752881762150845620, 3.54836043949067546959013704476, 4.57041588382976734943465349390, 5.97776496067737936530168155956, 6.65172095489332773595196796393, 7.999164874241700356366186399819, 8.360088928861532647193462840980, 9.099967237966268641525759865103, 10.03074935626406960154103357609
